Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals → CMB
Abstract view

# A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings

Published:2009-03-01
Printed: Mar 2009
• Jakob Cimpri\v{c}
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.
 Keywords: Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry
 MSC Classifications: 16W80 - Topological and ordered rings and modules [See also 06F25, 13Jxx] 46L05 - General theory of $C^*$-algebras 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22] 14P99 - None of the above, but in this section

 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2015 : https://cms.math.ca/